I’m curious as to how knowledgeable the Europeans and North Africans were of the size of the mongol empire that threatened at the time of its conquests through til it’s collapse. They would’ve known about China so would they have known that the same country that had conquered and ruled China was now threatening the holy lands and the Kieran Rus?
There are multiple accounts from European and Islamic travelers through the Mongol empire, all of which related the vast distances involved (for most, their whole point was to relay info back to Europe). My favorite is William of Rubruck(1253) but there are a number of famous accounts like those of John of Plano Carpini (1240s), Ibn Batuta (1350s) and yes, even Marco Polo (1300)
...and scholars already knew of the distances involved from the roman/han trading routes ...and before that, hellenistic kingdoms in Central Asia.
However, if you're asking to what extent people knew the information, that's difficult (do many people really appreciate the size of Eurasia even today?) Vatican scholars definitely had information and considering that the Mongols were in correspondence with a number of kingdoms (most famously France) it's reasonable to say that some people knew reasonably well.
...but it's at this point that I'll pull you up on your timeline. The Mongols only completely defeated Song China in 1279. This was well after the threats to Europe under ogedei khagan had subsided (1240s) and even after the Mongol defeat at Ayn Jalut (1260) Even before the defeat of song china, the Mongol empire descended into civil war and wasn't the expansionary threat it had once been.
Did Europeans know much about the extent of the Mongol empire when it first encountered it in the 1220s-1240s? No, which is evidenced by the fact that they sent people to compile info... But you need to remember that the empire wasn't in its final state when that threat was being posed.